Quadratic Interpolation and Rayleigh-Ritz Methods for Bifurcation Coefficients

TL;DR

This paper develops a method using quadratic interpolation and Rayleigh-Ritz approximation to accurately estimate bifurcation coefficients in nonlinear problems, ensuring convergence and providing sharp convergence rates with practical examples.

Contribution

It introduces a novel approach combining quadratic interpolation with Rayleigh-Ritz to improve convergence analysis of bifurcation coefficient estimations.

Findings

Convergence of approximations in strong norms is established.

Sharp convergence rate results are derived.

Practical examples demonstrate the method's effectiveness.

Abstract

In this article we study the estimation of bifurcation coefficients in nonlinear branching problems by means of Rayleigh-Ritz approximation to the eigenvectors of the corresponding linearized problem. It is essential that the approximations converge in a norm of sufficient strength to render the nonlinearities continuous. Quadratic interpolation between Hilbert spaces is used to seek sharp rate of convergence results for bifurcation coefficients. Examples from ordinary and partial differential problems are presented.